Antarctic Whale Project: Single Species

1 Abstract

2 Descriptive Statistics

2.1 Distance

2.2 Time

2.3 Velocity

2.4 Angles

3 Correlated random walk

Process Model

\[ d_{t} \sim T*d_{t-1} + Normal(0,\Sigma)\] \[ x_t = x_{t-1} + d_{t} \]

3.1 Parameters

For each individual:

\[\theta = \text{Mean turning angle}\] \[\gamma = \text{Move persistence} \]

For both behaviors process variance is: \[ \sigma_{latitude} = 0.1\] \[ \sigma_{longitude} = 0.1\]

3.2 Behavioral States

\[ \text{For each individual i}\] \[ Behavior_1 = \text{traveling}\] \[ Behavior_2 = \text{foraging}\]

\[ \alpha_{i,1,1} = \text{Probability of remaining traveling when traveling}\] \[\alpha_{i,2,1} = \text{Probability of switching from feeding to traveling}\]

\[\begin{matrix} \alpha_{i,1,1} & 1-\alpha_{i,1,1} \\ \alpha_{i,2,1} & 1-\alpha_{i,2,1} \\ \end{matrix} \]

3.3 Environment

Behavioral states are a function of local environmental conditions. The first environmental condition is ocean depth. I then build a function for preferential foraging in shallow waters.

It generally follows the form, conditional on behavior at t -1:

\[Behavior_t \sim Multinomial([\phi_{traveling},\phi_{foraging}])\] \[logit(\phi_{traveling}) = \alpha_{Behavior_{t-1}} + \beta_1 * Ocean_{y[t,]}\] \[logit(\phi_{foraging}) = \alpha_{Behavior_{t-1}} + \beta_2 * Ocean_{y[t,]}\]

3.4 Continious tracks

The transmitter will often go dark for 10 to 12 hours, due to weather, right in the middle of an otherwise good track. The model requires regular intervals to estimate the turning angles and temporal autocorrelation. As a track hits one of these walls, call it the end of a track, and begin a new track once the weather improves. We can remove any micro-tracks that are less than three days. Specify a duration, calculate the number of tracks and the number of removed points. Iteratively.

How did the filter change the extent of tracks?

sink(“Bayesian/Multi_RW.jags”) cat(" model{

#Constants
pi <- 3.141592653589

##argos observation error##
argos_prec[1:2,1:2] <- inverse(argos_sigma*argos_cov[,])

#Constructing the covariance matrix
argos_cov[1,1] <- 1
argos_cov[1,2] <- sqrt(argos_alpha) * rho
argos_cov[2,1] <- sqrt(argos_alpha) * rho
argos_cov[2,2] <- argos_alpha

for(i in 1:ind){
  for(g in 1:tracks[i]){

    ## Priors for first true location
    #for lat long
    y[i,g,1,1:2] ~ dmnorm(argos[i,g,1,1,1:2],argos_prec)

    #First movement - random walk.
    y[i,g,2,1:2] ~ dmnorm(y[i,g,1,1:2],iSigma)
    
    ###First Behavioral State###
    state[i,g,1] ~ dcat(lambda[]) ## assign state for first obs
    
    #Process Model for movement
    for(t in 2:(steps[i,g]-1)){
    
    #Behavioral State at time T
    logit(phi[i,g,t,1]) <- alpha[i,state[i,g,t-1]] + beta[i,state[i,g,t-1]] * ocean[i,g,t] + beta2[i,state[i,g,t-1]] * coast[i,g,t]
    phi[i,g,t,2] <- 1-phi[i,g,t,1]
    state[i,g,t] ~ dcat(phi[i,g,t,])
        
    #Turning covariate
    #Transition Matrix for turning angles
    T[i,g,t,1,1] <- cos(theta[state[i,g,t]])
    T[i,g,t,1,2] <- (-sin(theta[state[i,g,t]]))
    T[i,g,t,2,1] <- sin(theta[state[i,g,t]])
    T[i,g,t,2,2] <- cos(theta[state[i,g,t]])
    
    #Correlation in movement change
    d[i,g,t,1:2] <- y[i,g,t,] + gamma[state[i,g,t]] * T[i,g,t,,] %*% (y[i,g,t,1:2] - y[i,g,t-1,1:2])
    
    #Gaussian Displacement
    y[i,g,t+1,1:2] ~ dmnorm(d[i,g,t,1:2],iSigma)
  }

#Final behavior state
logit(phi[i,g,steps[i,g],1]) <- alpha[i,state[i,g,steps[i,g]-1]] + beta[i,state[i,g,steps[i,g]-1]] * ocean[i,g,steps[i,g]] + beta2[i,state[i,g,steps[i,g]-1]] * coast[i,g,steps[i,g]]
phi[i,g,steps[i,g],2] <- 1-phi[i,g,steps[i,g],1]
state[i,g,steps[i,g]] ~ dcat(phi[i,g,steps[i,g],])

##  Measurement equation - irregular observations
# loops over regular time intervals (t)    

for(t in 2:steps[i,g]){

# loops over observed locations within interval t
for(u in 1:idx[i,g,t]){ 
  zhat[i,g,t,u,1:2] <- (1-j[i,g,t,u]) * y[i,g,t-1,1:2] + j[i,g,t,u] * y[i,g,t,1:2]

    #for each lat and long
      #argos error
      argos[i,g,t,u,1:2] ~ dmnorm(zhat[i,g,t,u,1:2],argos_prec)
      }
    }
  }
}
###Priors###

#Process Variance
iSigma ~ dwish(R,2)
Sigma <- inverse(iSigma)

##Mean Angle
tmp[1] ~ dbeta(10, 10)
tmp[2] ~ dbeta(10, 10)

# prior for theta in 'traveling state'
theta[1] <- (2 * tmp[1] - 1) * pi

# prior for theta in 'foraging state'    
theta[2] <- (tmp[2] * pi * 2)

##Move persistance
# prior for gamma (autocorrelation parameter) in state 1
gamma[1] ~ dbeta(5,2)

# prior for gamma in state 2
gamma[2] ~ dbeta(2,5)

##Behavioral States
# Following lunn 2012 p85

#Hierarchical structure
#Intercepts
alpha_mu[1] ~ dnorm(0,0.386)
alpha_mu[2] ~ dnorm(0,0.386)

#Variance
alpha_tau[1] ~ dt(0,1,1)I(0,)
alpha_tau[2] ~ dt(0,1,1)I(0,)

#Slopes
## Ocean Depth
beta_mu[1] ~ dnorm(0,0.386)
beta_mu[2] ~ dnorm(0,0.386)

# Distance coast
beta2_mu[1] ~ dnorm(0,0.386)
beta2_mu[2] ~ dnorm(0,0.386)

#Variance
#Ocean
beta_tau[1] ~ dt(0,1,1)I(0,)
beta_tau[2] ~ dt(0,1,1)I(0,)

#Coast
beta2_tau[1] ~ dt(0,1,1)I(0,)
beta2_tau[2] ~ dt(0,1,1)I(0,)

#For each individual
for(i in 1:ind){
  # prob of being in state 1 at t, given in state 1 at t-1    
  #Individual Intercept
  alpha[i,1] ~ dnorm(alpha_mu[1],alpha_tau[1])
  
  #effect of ocean on traveling -> traveling
  beta[i,1] ~ dnorm(beta_mu[1],beta_tau[1])
  
  #effect of coast on traveling -> traveling
  beta2[i,1] ~ dnorm(beta2_mu[1],beta2_tau[1])

  #Prob of transition to state 1 given state 2 at t-1
  alpha[i,2] ~ dnorm(alpha_mu[2],alpha_tau[2])

  #effect of ocean on feeding -> traveling
  beta[i,2] ~ dnorm(beta_mu[2],beta_tau[2])

  #effect of dist to coast on feeding -> traveling
  beta2[i,2] ~ dnorm(beta_mu[2],beta_tau[2])

}

#Probability of behavior switching 
lambda[1] ~ dbeta(1,1)
lambda[2] <- 1 - lambda[1]

##Argos priors##
#longitudinal argos error
argos_sigma ~ dunif(0,10)

#latitidunal argos error
argos_alpha~dunif(0,10)

#correlation in argos error
rho ~ dunif(-1, 1)


}"
,fill=TRUE)

sink()

##      user    system   elapsed 
##    16.050     0.791 17316.957

3.5 Chains

3.5.1 Compare to priors

3.6 Prediction - environmental function

3.6.1 Ocean Depth

3.6.2 Distance to Coast

4 Behavioral Prediction

4.1 Spatial Prediction

4.1.1 Per Animal

## $`1`

## 
## $`2`

## 
## $`3`

## 
## $`4`

## 
## $`5`

## 
## $`6`

## 
## $`7`

## 
## $`8`

## 
## $`9`

4.2 Log Odds of Feeding

4.3 Autocorrelation in behavior

4.4 Behavioral description

4.5 Predicted behavior duration

4.6 Location of Behavior

Global Plotting

5 Krill Fishery

##             used  (Mb) gc trigger   (Mb)   max used   (Mb)
## Ncells   1556039  83.2   32145108 1716.8   50226733 2682.4
## Vcells 126534874 965.4  969011838 7393.0 1171835194 8940.4